Edits to OLG lecture (#110)

* misc

* Add olg to _toc in models

---------

Co-authored-by: mmcky <mamckay@gmail.com>

Author: jstac

in-work/OLG.md

@@ -28,6 +28,7 @@ parts:
   - file: schelling
   - file: solow
   - file: cobweb
+  - file: olg
 - caption: Other
   numbered: true
   chapters:

lectures/_toc.yml

@@ -0,0 +1,159 @@
+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.4
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
+
+# The Overlapping Generations Model
+
+In this lecture we study the overlapping generations (OLG) model.
+
+The dynamics of this model are quite similar to Solow-Swan growth model.
+
+At the same time, the OLG model adds an important new feature: the choice of
+how much to save is endogenous.
+
+To see why this is important, suppose, for example, that we are interested in
+predicting the effect of a new tax on long-run growth.
+
+We could add a tax to the Solow-Swan model and look at the change in the
+steady state.
+
+But this ignores something important: households will change their behavior
+when they face the new tax rate.
+
+Some might decide to save less, and some might decide to save more.
+
+Such changes can substantially alter the predictions of the model.
+
+Hence, if we care about accurate predictions, we should model the decision
+problems of the agents.
+
+In particular, households in the model should decide how much to save and how
+much to consume, given the environment that they face (technology, taxes,
+prices, etc.)
+
+The OLG model takes up this challenge.
+
+We will present a simple version of the OLG model that clarifies the decision
+problem of households and studies the implications for long run growth.
+
+
+## The Model
+
+We assume that
+
+- time is discrete, so that $t=0, 1, \ldots$ and
+- individuals born at time $t$ live for two periods: dates $t$ and $t + 1$.
+
+
+### Preferences
+
+Suppose that the utility functions take the familiar constant relative risk
+aversion (CRRA) form, given by:
+
+```{math}
+:label: eq_crra
+    U_t = \frac{c^1_t^{1-\gamma}-1}{1-\gamma} +
+           \beta \left( \frac{c^2_{t+1}^{1-\gamma}-1}{1-\gamma} \right )
+```
+
+Here
+
+- $\gamma$ is a parameter and $\beta \in (0, 1)$ is the discount factor
+- $c^1_t$ is time $t$ consumption of the individual born at time $t$
+- $c^2_t$ is time $t+1$ consumption of the same individual (born at time $t$)
+
+### Production
+
+For each integer $t \geq 0$, output $Y_t$ in period $t$ is given by
+
+$$
+    Y_t = F(K_t, L_t)
+$$
+
+Here $K_t$ is capital, $L_t$ is labor and $F$ is an aggregate production function.
+
+Without population growth, $L_t$ equals some constant $L$.
+
+### Prices
+
+Setting $k_t := K_t / L$, $f(k)=F(K, 1)$ and using homogeneity of degree one now yields:
+
+$$
+    1 + r_t = R_t = f'(k_t)
+$$
+
+The gross rate of return to saving is equal to the rental rate of capital.
+
+The wage rate is given by
+
+$$
+    w_t = f(k_t) - k_t f'(k_t)
+$$
+
+
+
+### Equilibrium
+
+Savings by an individual of generation $t$, $s_t$, is determined as a
+solution to:
+
+$$
+    \begin{aligned}
+    \max_{c^1_t, c^2_{t+1}, s_t} \ & u(c^1_t) + \beta u(c^2_{t+1}) \\
+    \mbox{subject to } \ & c^1_t + s_t \le w_t \\
+                         & c^2_{t+1} \le R_{t+1}s_t\\
+    \end{aligned}
+$$
+
+The second constraint incorporates notion that individuals only spend
+money on their own end of life consumption.
+
+Solving for consumption and thus for savings,
+
+$$
+    s_t = s(w_t, R_{t+1})
+$$
+
+Total savings in the economy will be equal to:
+
+$$
+    S_t = s_t L_t
+$$
+
+
+
+### Dynamics
+
+We assume a closed economy, so domestic investment equals aggregate domestic
+saving. Therefore, we have
+
+$$
+    K_{t+1} = L_t s(w_t, R_{t+1})
+$$
+
+Setting $k_t := K_t / L_t$, where $L_{t+1} = (1 + n) L_t,$ and using homogeneity of degree one now yields:
+
+```{math}
+:label: k_dyms
+    k_t = \frac{s(w_t, R_{t+1})}{1 + n}
+```
+
+
+
+A steady state is given by a solution to this equation such that
+$k_{t+1} = k_t = k^*$, i.e,
+
+```{math}
+:label: k_star
+    k^* = \frac{s(f(k^*)-k^*f'(k^*), f'(k^*))}{1+n}
+```